Course Outline of BS Mathematics in UoK

Department of mathematical sciences University of Karachi

Depart iences

ment Mathematical Sc

University of Karachi

PROGRAMS

1. BS in Pure and Applied Mathematics

ics

3. atics

2. MS in Pure and Applied Mathematm

4.

Ph.D. in Pure and Applied Mathe

5.

Certificate Course in Mathematics

Diploma Course in Mathematics

6. Diploma in Information Technology

LIST OF FIRST YEAR COURSES

FIRST SEMES

TER

Course Title

Sr.No. Course No. Credit Hours 1 MATH 301 Algebra and Calculus ‐ I 3

2 301 Sub ‐ I 3

3 301 Sub ‐ II 3

4 3 00.1 ( I.S ) Islamic Studies ( Compulsory ) 3

5 300.1 ( E ) English ( Compulsory ) 3

Total Credit Hours 15

SECOND SEMESTER

Course Title

Sr.No. Course No. Credit Hours 1 MATH 302 Algebra and Calculus ‐ II 3

2 302 Sub ‐ I 3

3 302 Sub ‐ II 3

4 3 00.2 ( P.S ) Pakistan Studies ( Compulsory ) 3

5 300.2 ( U ) Urdu / Sindhi / NaturaCompulsory )

l Science ( 3


Total Number of Courses = 10

Total Number of Credit hours = 30

LIST OF SECOND YEAR COURSES

THIRD SEMES

TER

Sr.No.

Course No.

Course Title Credi s t Hour

1 MATH 401 Mechanics and Geometry ‐ I 3

2 MATH 403 Data Processing and Programming ‐ I 2 1 + 3 401 Sub ‐ I 3

4 401 Sub ‐ II 3

5 400.1 Biology (Costudents of

mpulsory ) ( for the Physical Sciences ) 3

6 400.1 ( E ) English ‐ II 3


FORTH SEMES

TER

Sr.No.

Course No.

Course Title Credi s t Hour

1 MATH 402 Mechanics and Geometry ‐ II 3

2 MATH 404 Data Processing and Programming ‐ II 2 1 + 3 402 Sub ‐ I 3

4 402 Sub ‐ II 3

5 402.1 Biology (Compulsory ) ( for the students of Physical Science ) 3

6 400.1 ( C.S ) Computer Applications ( u Comp lsory ) 3



otal Number of Credit hours = 36 T

LIST OF THIRD YEAR COURSES

e

FIFT SEMEST

H

ER

Course No. Course Ti

Sr.No. tl Credit Hours 1 MAT H 501 Analysis ‐ I 3

2 MATH 503 Algebra ‐ I 3

3 MATH 505 Numerical Analysis ‐ I 2 1 +4 MATH 507 Applicable Differential Geometry ‐ I 3

5 MATH 509 Methods of Mathematical Physics ‐ I 3

6 Introduction to Social Sciences 3


SIXT SEMEST

H

ER

Course No. Course Ti

Sr.No. tl e Credit Hours 1 MATH 502 Analysis ‐ II 3

2 MATH 504 Algebra ‐ II 3

3 MATH 506 Numerical Analysis ‐ II 2 1 +4 MATH 508 Applicable Differential Geometry ‐ II 3

5 MATH 510 Methods of Mathematical Physics ‐ II 3

6 Communication Skills 3



otal Number of Credit hours = 36 T

LIST O SESF FOURTH YEAR COUR

S

( PURE MATHEMATICS ) OMPULSORY COURSES C

Sr.No.

Course No.

Course Title

1

MATH 601

Abstract Algebra

2

3

MATH 605

MATH 647

Measure Theory

Proje

ctive Geometry

( APPLIE )

S

D MATHEMATICS

COMP R UR E

e

ULSO Y CO

Course Titl

Sr.No.

Course No.

1

MATH 607

APPLIED ALGEBRA

2

3

MATH 609

MATH 655

BIO‐‐MATHEMATI

FLUID DYN

CS

AMICS

TIONAL COURSES

OP

Sr.No.

Course No.

Course Title

APPLIED ALGEBRA

1

MATH 607

(P)

2

MATH 611

FUNCTIONAL ANALYSIS

SIS

(P/A)

3

MATH 631

APPLIED NUMERICAL ANALY

CS

(P/A)

4

5

MATH 633

MATH 645

MATHEMATICAL STATISTI

OPERATIONAL RESEARCH

(P/A)

(P/A)

6

7

MATH 651

MATH 661

CLASSICAL MECHANIC

ELECTROMAGNETISM

S (P/A)

(P/A)

ATICS, DEPARTMENT OF MATHEM

UNIVERSITY OF KARACHI,

SY BU S. FOUR YEAR PROGRAM LLA S FOR B.

FIRST SEMESTER

BRA AND CALCULUS ‐ I MATH 301: ALGE

ents: Course cont

SECTION A:

NUMBER SYSTEM: Real and complex number systems, De Moiver’s theorem with applications, exponential, trigonometric, hyperbolic, logarithmic, inverse hyperbolic and inverse circular functions.

INFINITE SERIES: Sequences, limits and bounds of sequences, infinite series, basic comparison test, limit comparison test, integral, ratio and root tests, alternating series,

d conditional convergence. absolute an

SECTION B:

SET THEORY: Binary relations, functions and their graphs, composition of functions.

GROUP THEORY: Groups and their properties, subgroups, order of a group, cyclic groups, cosets, Lagrange’s theorem, permutation groups, rings, fields, vector spaces, subspaces, linear combinations and spanning set, linear dependence and basis, dimension, linear

ions. transformat

SECTION C:

DIFFERENTIAL CALCULUS: Bounds, limits and continuity, properties of continuous functions, derivatives, Leibnitz and Rolle’s theorems, Lagrange’s and Cauchy’s mean value

theorems, generalized mean value theorems, indeterminate forms, Taylor’s and series. Maclaurin’s

SECTION D:

INTEGRAL CALCULUS: Anti‐derivatives, techniques of integration, Riemann integral, properties of definite integrals, mean value theorem, reduction formulae, improper integrals and Beta and gamma integrals.

FOURIER SERIES: Periodic function, periodic extensions, even and odd functions, Fourier coefficients, expansion of functions in Fourier series, functions with arbitrary periods,

e series. Fourier sine and cosin

Books Recommended:

1. Yousuf, S. M., Mathematical Methods, Fourth Edition, Ilmi Kitab Khana, Lahore, 2003.

2. Calvert, J. and Voxman, W., Finite Mathematics, McGraw Hill, N.Y., 1994.

3. Kreyszig, E., Advanced Engineering Mathematics, Ninth Edition, John Wiley, 2005.

4. Jain, M. K., Iyengam, S.R.K. and Jain, R.K., Numerical Methods For Scientific and Engineering Computations, Six Edition, Wiley Esastern Ltd., 1991.

5. Anton, H., Elementary Linear Algebra, Eight Edition, John Wiley, 1997.

6. Thorde, J. A. and Kumpel, P.G., Elementary Linear Algebra, Saunders College Publishers, N.Y., 1984.

7. Talpur, N. M., Calculus and Analytic Geometry, Ferozesons, 1971.

8. Thomas and Finney, Calculus and Analytic Geometry, Addision Wesley, 2005.

9. Boyce, W. E. and Prima, R. C., Elementary Differential Equations and Boundary Value Problems, John Wiley, 1992.

10. Flus, R., Calculus and Analytic Geometry, Prindle, Weber and Schmidt, Boston, Mass, 1983.

11. Swokowski, E. W., Calculus and analytic geometry, Prindle, Weber and Schmidt Bosten, Mass, 2000.

12. Adler, F. R., Modeling the Dynamics of Life Calculus and Probability for Life Science, Second Edition, Thomson Brooks / Cole, 2005.

13. Sharma, S. C., Complex Variable, First Edition, Discovery Publishing House, New Delhi, 2007.

14. Sharma, A.K., Power Series, First Edition, Discovery Publishing House, New Delhi, 2007.

15. Jain, R. K. and Iyengar, S.R.K., Advanced Engineering Mathematics, Third Edition, Narosa Publishing House, New Delhi, 2007.

16. O’Neil, P. V., Advanced Engineering Mathematics, Fifth Edition, 2003

17. Mathews, J. H. and Howell, R. W., Complex Analysis for Mathematics and Engineering, Fifth Edition, Jones and Bartlett Publishers, Boston, 2006

18. Steward, Precalculus Mathematics for Calculus, Forth Edition, with CD, Brooks Cole, 2002.

19. Kishan H., Differential Calculus, Atlantic Publishers and Distributors Pvt. Ltd., 2007.

SE

BRA AND CALCULUS ‐ II

COND SEMESTER

MATH 302: ALGE

ents: Course cont

SECTION A:

MATRICES: Elementary row operations, echelon and reduced echelon forms, inverse, rank and normal form of a matrix, matrix of linear transformation, partitioning of a matrix.

DETERMINANTS: Axiomatic definition of a determinant, determinant as sum of product of Aelements, djoint and inverse of a matrix.

SYSTEMS OF LINEAR EQUATIONS: Gaussian elimination and Gauss‐Jordan methods, le, consistent and inconsistent systems. Cramer’s ru

SECTION B:

EQUATIONS: Solutions of cubic and biquadratic equations, numerical solution of equations, l .Newton‐Raphson, regula fa si and bisection methods

PROBABILITY: Axioms of Probability, conditional probability, discrete and continuous iables, probability distributions, binomial, Poisson and normal distributions. random var

SECTION C:

DIFFERENTIAL EQUATIONS I: Differential equations, formation and solution, equations of first order, initial and boundary value problems, various methods of solving first order differential equations: Separable, Exact & Homogeneous equation, integration factor and orthogonal trajectories. Non‐Linear First Order Equations, Envelopes and Singular solutions.

SECTION D:

DIFFERENTIAL EQUATIONS II: Higher order Homogeneous Differential equations with constant coefficients, superposition of solutions, Cauchy‐Euler’s equations, systems of two

genous equations, nonlinear equations. first order linear homo

Books Recommended:

ahore, 2003. 1. Yousuf, S. M., Mathematical Methods, Fourth Edition, Ilmi Kitab Khana, L

2. Calvert, J. and Voxman, W., Finite Mathematics, McGraw Hill, N.Y., 1994.

3. Kreyszig, E., Advanced Engineering Mathematics, Ninth Edition, John Wiley, 2005.

4. Jain, M. K., Iyengam, S. R. K. and Jain, R.K., Numerical Methods For Scientific and Engineering Computations, Six Edition, Wiley Esastern Ltd, 1991.

5. Anton, H., Elementary Linear Algebra, Eight Edition, John Wiley, 1997.

6. Thorde, J. A. and Kumpel, P.G., Elementary Linear Algebra, Saunders College Publishers, N.Y., 1984.

7. Talpur, N. M., Calculus and Analytic Geometry, Ferozesons, 1971.

8. Thomas and Finney, Calculus and Analytic Geometry, Addision Wesley, 2005.

9. Boyce, W. E. and Prima, R. C., Elementary Differential Equations and Boundary Value Problems, John Wiley, 1992.

10. Flus, R., Calculus and Analytic Geometry, Prindle, Weber and Schmidt, Boston, Mass, 1983.

11. Swokowski, E. W., Calculus and Analytic Geometry, Prindle, Weber and Schmidt Bosten, Mass, 2000.

12. Adler, F. R., Modeling the Dynamics of Life Calculus and Probability for Life Science, Second Edition, Thomson Brooks / Cole, 2005.

13. Sharma, J. N., Numerical Methods for Engineers and Scientists, Second Edition, Narosa Publishing House, New Delhi, 2007.

14. Birkhoff, G. and Rota, G. C. ,Ordinary Differential Equations, Forth Edition, John Wiley and Sons, New York, 1989.

15. Sharma, A. K., Linear Transformations, First Edition, Discovery Publishing House, New Delhi, 2007.

16. Jain, R. K. and Iyengar, S. R. K., Advanced Engineering Mathematics, Third Edition, Narosa Publishing House, New Delhi, 2007.

17. O’Neil, P. V., Advanced Engineering Mathematics, Fifth Edition, 2003

18. Steward, Precalculus Mathematics for Calculus, Forth Edition, with CD, Brooks Cole, 2002.

19. Kishan H., Differential Calculus, Atlantic Publishers and Distributors Pvt. Ltd., 2007

THIR

ANICS AND GEOMETRY ‐ I

D SEMESTE R

MATH 401: MECH

ents: Course cont

SECTION A:

VECTOR ANALYSIS: Differentiation and integration of vectors, scalar and vector fields, gradient, divergence and curl, line, surface and volume integrals, theorems of Green, Gauss,

out proofs),and their applications. Stoke (with

SECTION B:

STATICS: Composition of forces, equilibrium problems, moments and couples, centre of avity, friction, virtual work, flexible cables, catenaries. mass and gr

SECTION C:

PLANE CURVES: Curves in Cartesian plane, parametric representation, polar coordinates, tangents and normals, polar equation of a conic, Pedal equation, Change of axes, general equation of second degree. Extreme values, singular points, asymptotes, curve tracing,

insic equation, curvature, areas in rectangular and polar coordinates. length of arc, intr

:SECTION D

ANALYTIC GEOMETRY IN THREE DIMENSIONS: Direction cosine and direction ratios, equations of a line, angle between two lines, distance of a point from a line, shortest distance between two lines, equation of a plane, angle between planes, area of a triangle

and volume of tetrahedron, spherical and cylindrical polar coordinates, surfaces, metry, quadric surfaces, sphere, surface of revolution, ruled surfaces. intercepts, traces, sym

Books Recommended:

1. Ghori, Q. K., (Ed.), Introduction to Mechanics, West Pakistan Publishing Co. 1971.

, 1970. 2. Synge, J. L. and Griffith, Principles of Mechanics, McGraw Hill, Kogakusha

, Chichest, 1977. 3. Chorlton, F., Vector and Tensor Methods, Ellis Horwood

4. Chorlton, F., Mechanics, Van Nostrand, Reinhold, 1970.

5. Parakash, N., Differential Geometry, McGraw Hill, 1990.

6. Goetz, A., Introduction to Differential Geometry, Addison Wesley, 1970.

7. Sharma, S. C., Complex Integration, First Edition, Discovery Publishing House, New Delhi, 2007.

PROCESSING AND PROGRAMMING ‐ I ( 2 + 1 ) MATH 403: DATA

Course contents:

Digital logic: Basic computer mathematics, binary, hexadecimal and other arithmetic, symbolic logic, logic circuits and gates, codes, encoding and decoding. Boolean algebras, Karnough maps, arithmetic unit, control unit, memory/storage, input unit, output device.

Computer based communication, networking, fax/modem, electronic mail. Disk operating systems, working with DOS. Information / data processing concepts, data processing cycle, data processing operations. Algorithm design technique. Programming in Visual Basic,

ing in Visual Basic. problem solv

PRACTICALS.

ands. 1. Using DOS Comm

2. Using Windows.

Basic programs. 3. Running Visual

Books Recommended:

1. Aho, A., The Design and Analysis of Computer Algorithms, Addison Wesley, Reading Mass, 1974.

2. Burgard, M. J., Dos Unix Networking and Internetworking, J. Wiley, New York, 1994. 3. eDate, C. J., An Introduction to Database Syst ms, Fourth edition, Addison Wesley,

Reading Mass, 1986. 4. Horowitz, E., and Sahni, S., Fundamentals of Computer Algorithms, Computer

Science Press, Potomac, Maryland, 1978. 5. , s eUllman J. D., Principles of Databa e Systems, Comput r Science Press, Potomac,

Maryland, 1980. 6. . , C iWeiss, M A. Data Structures and Algorithm Analysis, Benjamin umm ngs, New

York, 1992. 7. ctures Fortran 77, Little / Brown, Mashaw, B., Programming Byte by Byte Stru

.8.

Boston, 1983 Rudd, A., Mastering C, J. Wiley, New York, 1994.

9. Crandall, R. E., Mathematica for the Sciences, Addison Wesley, Redwood City, California, 1991.

10. and Glynn, r w AGray, T., J., Explo ing Mathematics ith Mathematica, ddison Wesley, Redwood City, California, 1991.

11. Maeder, R., Programming in Mathematica, Addison Wesley, Redwood City, California, 1991.

12. Skiena, S., Implementing Discrete Mathematics: Combinatories and Graph Theory with Mathematica, Addison Wesley, Redwood City, California 1990.

13. , M h : tWolfram S., at ematica A System for Doing Mathema ics by Computer, second edition, Addison Wesley, Redwood City, California, 1991.

14. crocomputer Artwick, B. A., Applied Concepts in Mi Graphics, Prentice Hall, Englewood Cliffs, New Jersey, 1984.

15. es / Cole Demel, J. T., and Miller, M. J., Introduction to Computer Graphics, Brook

16. Engineering Division, Monterey1984. Escher, M. C., The Graphic Work of M.C. Esher, Ballantine, New York, 1971.

17. Foley, J. D. and Van D. A., Fundamentals of Interactive Computer Graphics, Addison Wesley, Redwood City, Calif ornia, 1982.

FOURT

ANICS AND GEOMETRY ‐ II

H EMESTER S

MATH 402: MECH

ents: Course cont

SECTION A:

DYNAMICS I: Galilean‐Newtonian principle, inertial frames, Galilean transformations, kinematics, rectilinear motion with variable accelerations, simple harmonic motion,

dynamics, principles of energy and momentum. methods of

SECTION B:

DYNAMICS II: Motion of a projectile, orbital motion, moment of inertia, motion of a rigid body, plane impulsive motion Compound pendulum.

SECTION C:

DIFFERENTIAL GEOMETRY: Simple arcs and curves in three dimension and their parametric representation, the arc length, the natural Parameterization, contacts (of order up to two) of curves and a surface, osculating plane, Frenet trihedron and Frenet formulae, curvature and torsion of curves, surfaces in space, curvilinear coordinates, implicit equation of surface, tangent plane, curves on surfaces and tangent vector, angle between

surface, first and second fundamental forms on a surface. curves on a

SECTION D:

MULTIVARIATE CALCULUS: Partial derivatives, geometrical meaning, equation of tangent plane and normal to surfaces, chain rule, approximation with the help of differentials, homogeneous functions, Euler’s theorem, evaluation of simple double and triple integrals,

eas of solids of revolutions. volume and surface ar

Books Recommended:

1. . Ghori, Q. K., (Ed.), Introduction to Mechanics, West Pakistan Publishing C Hill, Kogakusha

3. , Chichest, 1977.

o. 19712. , 1970. Synge, J. L. and Griffith, Principles of Mechanics, McGraw

4. Chorlton, F., Vector and Tensor Methods, Ellis Horwood

5. Chorlton, F., Mechanics, Van Nostrand, Reinhold, 1970.

6. Parakash, N., Differential Geometry, McGraw Hill, 1990. Goetz, A., Introduction to Differential Geometry, Addison Wesley, 1970.


8. Meirovitch L., Methods of Analytical Dynamics, First edition, McGraw Hill, New York, 2007.

PROCESSING & PROGRAMMING ‐ II ( 2 + 1 ) MATH 404: DATA

Course contents:

Design technique, algorithm analysis, complexity of algorithm, randomized algorithm and simulation concept. General features of Fortran/ C/ Turbo C; operators, statements, loops, functions, pointers, arrays, structures and files. Data manipulation in lists, linked lists, searching, sorting, and duplicating; tree algorithms. File concept, different access modes, print control, standard functions, user defined functions and subroutines.

omputing using Mathematica/ Matlab/ Maple. Numerical c

PRARICALS:

1. . Programming in Fortran/ C/ Turbo Catica/ Matlab/ Maple. 2. Use of Mathem

Books Recommended:

1. g Aho, A., The Design and Analysis of Computer Algorithms, Addison Wesley, Readin

2. Mass, 1974. Burgard, M. J., Dos Unix Networking and Internetworking, J. Wiley, New York, 1994.

3. Date, C. J., An Introduction to Database Systems, Fourth edition, Addison Wesley, Reading Mass, 1986.

4. Horowitz, E. and Sahni, S., Fundamentals of Computer Algorithms, Computer Science Press, Potomac, Maryland, 1978.

5. , s eUllman J. D., Principles of Databa e Systems, Comput r Science Press, Potomac, Maryland, 1980.

6. . , C iWeiss, M A. Data Structures and Algorithm Analysis, Benjamin umm ngs, New York, 1992.

7. res Fortran 77, Little / Brown, Mashaw, B., Programming Byte by Byte Structu.

8. .Boston, 1983 Rudd, A., Mastering C, John Wiley, New York, 1994

9. Crandall, R. E., Mathematica for the Sciences, Addison Wesley, Redwood City, California, 1991.

10. and Glynn, r w AGray, T., J., Explo ing Mathematics ith Mathematica, ddison Wesley, Redwood City, California, 1991.

11. Maeder, R., Programming in Mathematica, Addison Wesley, Redwood City, California, 1991.

12. Skiena, S., Implementing Discrete Mathematics: Combinatories and Graph Theory with Mathematica, Addison Wesley, Redwood City, California 1990.

13. , M h : tWolfram S., at ematica A System for Doing Mathema ics by Computer, second edition, Addison Wesley, Redwood City, California, 1991.

14. crocomputer Artwick, B. A., Applied Concepts in Mi Graphics, Prentice Hall, Englewood Cliffs, New Jersey, 1984.

15. es / Cole Demel, J. T., and Miller, M. J., Introduction to Computer Graphics, Brook

16. Engineering Division, Monterey1984. Escher, M. C., The Graphic Work of M.C. Esher, Ballantine, New York, 1971.

17. Foley, J. D., and Van D., A., Fundamentals of Interactive Computer Graphics, Addison Wesley, Redwood City, California, 19.

FIFTH SEMESTER

MATH 501: ANALYSIS ‐ I

Course contents:

Algebra of sets; Partition and Equivalent classes, partially ordered sets and Axiom of Choice. Canonical decomposition of functions. Euclidean metric spaces (n ≥1). Convergence of sequences. Completeness. Functions of several real variables; their continuity and differentiability. Implicit and Inverse function Theorems. Taylor’s Theorem. Jacobians and functional dependence. Taylor’s Theorem (several variables). Maxima, Minima; Legrange’s method of undermined multipliers. Riemann and Riemann‐Stieltijes integrals. Differentiation under integral sign.

Books Recommended:

1. Apostol, T. M., Mathematical Analysis, Addison Wesley, 1978.

3. 2. Kaplan, W., Advanced Calculus, Addison Wesley, 1965.

Rudin, W., Principles of Mathematical Analysis, Mc Graw Hill, 1976. rk, 1983. 4. Taylor, A. E. and Mann, W. R., Advanced Calculus, C/M Wiley, New Yo

5. 5. Churchil, R. V., Complex Variables and Applications, Mc Graw Hill, 200

6. Paliouras, F. B., Complex Variables, Collier McMillan, New York, 1975.

Elements of Complex Variables, Holt, Rinechart and Winston, 7. Pennesi, L. L.,

New York, 1967.

8. Mathews, J. H., and Howell, R. W., Complex Analysis for Mathematics and Engineering, Fifth Edition, Jones and Bartlett Publishers, Boston, 2006

9. Jeffrey, A., Complex Analysis and Application, Second Edition, Chapman and Hall/CRC, New York, 2006


BRA ‐ I MATH 503: ALGE

Course contents:

Vector spaces: Definition and basic properties, Subspaces, Linear independence, linear combination and span. Basis and dimension change of basis. Orthogonal bases and projection in Rn. Inner product spaces. Linear transformations: Definition and examples. Properties of linear transformations. Range and Kernel. The rank and nullity of a matrix. The matrix representation of a linear transformation. Isomorphism, isometrics and their applications. Eigenvalues, eigenvectors and canonical forms: Eigenvalues, and eigenvectors, a model of population growth, similar matrices and diagonalisation, symmetric matrices and orthogonal diagonalisation. Quadratic forms. Matrix differential equations. The theorems of Cayley Hamilton and Gershgorin. Numerical methods: The error in numerical computations. Solving linear systems I: Gaussian elimination with pivoting. Solving linear

ethods. Computing eigenvalues and eigenvectors. systems II: Iterative m

Books Recommended:

1. Stanley, I., Grossman, Applied Linear Algebra, Second Edition, Wadsworth Publishing Co., California, 1984.

2. Stroud, K. A., Linear Algebra: Theory and Application, Stanley Thornes Publishers Ltd., 1978.

3. aticians, Graham, A., Matrix Theory and Applications for Engineers and MathemHalsted University, Ellis Horwood Ltd.,1980

4. Graham, A., Nonnegative Matrices and Applications for Engineers and Mathematicians, Halsted University, Ellis Horwood Ltd., 1987.

5. Lipschutz, S., Essential Computer Mathematics, Mc Graw Hill Inc., 1982.

6. Lennox, S. C., Chadwick, M., Computer Mathematics for Applied Scientists, Second Edition,

Heinemann Educational Books Ltd., London, 1985.

7. Garding and Tambour, Algebra and Switching Circuits, Mc Graw Hill 1988.

Hill 1978. 8. Mendelson, E., Boolean Algebra and Switching Circuits, Mc Graw

9. Halmon, P. R., Lectures on Boolean Algebra, Van Nostrand, 1963.

10. Sharma, A. K., Linear Transformations, First Edition, Discovery Publishing House, New Delhi, 2007.


ERICAL ANALYSIS – I (2 + 1) MATH 505: NUM

Course contents:

Errors Analysis: relative and absolute errors, percentage error, propagation. Root Finding Methods: Non‐linear equations in one unknown; Newton’s method, Secant method, Bisection method, Fixed Point Iteration method, Regula‐False Method. Polynomial Equations; Quotient Difference algorithm, Horner’s method Bairstow’s method. Systems of Equations: Linear Systems; Gaussian Elimination, Gauss‐Jacobi and Gauss‐Seidel iterative methods for diagonally dominant systems. III conditioned systems, Norms, condition numbers and errors in solution. Newton’s method for systems of Non‐Linear equations. Interpolation and Curve fitting: Development of polynomials for a given set of points. Lagrange polynomials, Newton’s Divided Difference Interpolation Polynomial for unevenly spaced data, Newton’s Forward Difference Interpolating Polynomial for evenly space data. Algorithm for developing a Cubic Spline. Least Square Approximations for fitting first, second and nth degree polynomials for a given set of data. Introduction to Bezier Curves and B‐Spline Curves. Numerical Differentiation: Differentiation using divided difference and forward difference Tables. Higher order derivatives, central difference formulas for derivatives of different order. Numerical Integration: Newton‐Cotes techniques for Numerical Integration and its use for developing Trapezoidal Rule,

rules. Gaussian Quadrature and Adaptive Integration. Simpson’s 1/3 and 3/8

Books Recommended:

1. Allen, III, M .B. and Isaacson, E. L., Numerical Analysis for Applied Sciences (Pure Sons and Applied Mathematics A. Willy‐interscience series texts), John Wiley and

Inc. N.Y. 1998 2. Jain, M. K., Iyengar, S. R. K. and Jain R. K., Computational methods for Partial

Differential Equations. Wiley Eastern Limited, New Delhi, 1991. 3. Jain, M. K., Iyengar, S. R. K. and Jain, R. K.: Numerical Methods for Scientific and

Engineering Computations. Wiley Eastern Limited, New Delhi, 1991. 4. Atkinson, K. E., An Introduction to Numerical Analysis, John Wiley and Sons, N.Y.,

1989. 5. Hager, W. W., Applied Numerical Linear Algebra, Prentice Hall International Inc.

Toronto, Canada, 1995. 6. Book Chapra, S. C. and Canale, R. P., Numerical Methods for Engineers, Mc Graw Hill

Co. Toronto, 2000. 7. Mathews, J. H. Numerical Methods for Mathematics, Science and Engineering,

Pentice Hall International Inc, N.J., 1984. 8. b. Gerald, C. F. and Patric, O.W., Applied Numerical Analysis, Addison Wesley Pu

9. Com., 1984. King, J. T., Introduction to Numerical Computation, Mc Graw Hill, N. Y., 1984.

10. Vendergraft, J. S., Introduction to Numerical Computation, Academic Press, NewYork, 1983.



13. Griffiths, D. V. and Smith, I. M., Numerical Methods for Engineers, Second Edition, Chapman and Hall/CRC, New York, 2006.

14. ist, Chapra, S. C., Applied Numerical Method, with MATLAB: For Engineers and ScientSecond Edition, Tufts University, McGraw Hill, 2007.

15. Karris S. T., Numerical Analysis using MATLAB and Excel, Third Edition, Orchant Pub., 2007.

16. Patil P. B., and Verma U. P., Numerical Computational Methods, Narosa Publication House, 2006.

ICABLE DIFFERENTIAL GEOMETRY ‐ I MATH 507: APPL

Course contents:

Notation, conventions and recapitulations of vector space theory. Affine spaces and subspaces, hyperplanes. Affine coordinate transformations. Affine maps. Smooth curves and functions on affine spaces. Tangent vectors; directional derivatives and derivations. Tangent space. Transition to Euclidean space and Frenet‐Serret formulae. Cotangent space; co‐vectors or 1‐forms. Curvilinear coordinates; coordinate transformations. Induced maps. Parallelism and covariant derivatives. Vector and co‐vector fields; lie derivative. General tensor: tensor algebra. Construction of new tensors from given tensor. Exterior algebra. Tensor fields and form fields. Calculus of forms: exterior derivatives. Lie derivatives. Closed and exact forms.

Books Recommended:

1. Crampin, M. and Pirani, F. A. E., Applicable Differential Geometry, CUP, 1986. Goetz, A., Introduction to Differential Geometry; Addison Wesley, 1970.

3. ntice Hall Inc., 2.

Milman, R. and Parker, G., Elements of Differential Geometry; Pre

4. 1977.

5. O’ Neill, B., Elementary Differential Geometry; Academic Press, 1995. Chorlton, F., Vector and Tensor Methods, Ellis Horwood, 1976.


ODS OF MATHEMATICAL PHYSICS ‐ I MATH 509: METH

Course contents:

Classification of differential equations and Solutions. Linear differential equations and superposition principle. Boundary value and initial value problems. Dynamical system; their analysis and control. Existence, uniqueness and stability of solutions. Function spaces; orthogonal sets of functions and generalized Fourier series. Strum‐Liouville Theory. Fourier series, integrals, transforms and applications. Linear ordinary differential equations of order n>1 (choose n=3). Existence and uniqueness theorem (statement and application only).Wronskian and fundamental sets of solutions. Methods of solution reduction of order, undetermined coefficients, variation of parameter and Green’s function. Power series solution. Legendre and Bessel’s equations. Properties of Legendre

l’s functions. polynomials and Besse

Books Recommended:

1. Boyce, W. E. and De Prima, R. C., Elementary Differential Equations and Boundary Value Problems; Fifth Edition, Wiley, New, 1992.

2. and W., Churchil, R. V. Brown, J. Fourier Series and Boundary Value Problems Third Edition, Mc Graw Hill Kogakusha, Tokyo, 1978.

3. Finnery, R. L. and Ostberg, D. R., Elementary Differential Equations with Linear Algebra Addison Wesley, Reading, Mass. 1976.

4. Rainville, E. D. and Bedient, P. E., Elementary Differential Equations, Seventh Edition , Macmillan, New York,1989

5. CoLeighton, W., First urse in Ordinary Differential Equations, Wadsworth Publishing Co. Belmont, California, 1981.

6. Arrowsmith, D. K. and Place, C. M., Ordinary Differential Equations, Chapman and Hall, 1982.

7. ns, New Barelli, R. L. and Coleman, C. S., Differential Equations, John Wiley and SoYork, 2004.

8. Humi, M. and Miller, W.B., Boundary Value Problem and Partial Differential Equations. PWS‐Kent Publishing Co., Boston, 1992.

9. Raisinghania, Ordinary and Partial Differential Equations, Chand Co., New Delhi, 2007.

10. Kevorkian, J., Partial Differential Equations, 2nd edition, Springer, 1999. Zwillinger D., Handbook of Differential Equations, AK Peters, 1992.

12. th 11.

Birkhoff, G. and Rota, G. C., Cengage Learning, Ordinary Differential Equations, ForEdition, John Wiley and Sons, New York, 1989.


14. O’Neil, P. V., Advanced Engineering Mathematics, Fifth Edition, Cengage Learning, 2003.

15. th Zill D. G. and Cullen M. R., Differential Equation with Boundary Value Problems, Fif

16. 005. Edition, Loyala Marymond Uni., Brooks Cole Pub., 2001. Kreyszig, E., Advanced Engineering Mathematics, Ninth Edition, John Willey, 2

17. ol Pandey R. K., Partial Differential Equation in several complex variable, AnmPublication Pvt. Ltd., 2008.

18. Pandey R. K., Partial Differential Equation in complex variable and integral transforms, Anmol Publication Pvt. Ltd., 2008.

19. O’Neil P. V., Beginning Partial Differential Equations, Second Edition, Wiley Interscience Pub., John Wiley an 0 8. d Sons Inc., 2 0

SIXTH SEMESTE R

MATH 502: ANALYSIS ‐ II

Course contents:

Line and surface integrals. Theorems of Gauss, Green and Stokes and their applications. Uniform and absolute convergence of sequences and series of functions. Uniform convergence and continuity. Term by term differentiation and integration. Improper integrals and their convergence; their absolute and uniform convergence. Functions of a complex variable. Analytic functions; power series, Cauchy’s Theorem and integral formulas. Singularities and branch points. Taylor’s and Laurent series. Resident theorem

ns conformal mapping. and contour integratio

Books Recommended:

1. Apostol, T. M., Mathematical Analysis, Addison Wesley, 1978.

3. 2. Kaplan, W., Advanced Calculus, Addison Wesley, 1965.

Rudin, W., Principles of Mathematical Analysis, Mc Graw Hill, 1976. rk, 1983. 4. Taylor, A. E. and Mann, W. R., Advanced Calculus, C/m Wiley, New Yo

5. 5. Churchil, R. V., Complex Variables and Applications, Mc Graw Hill, 200

6. Paliouras, F. B., Complex Variables, Collier McMillan, New York, 1975.

Elements of Complex Variables, Holt, Rinechart and Winston, 7. Pennesi, L.L.,

New York, 1967,

8. Mathews, J. H., and Howell, R. W., Complex Analysis for Mathematics and Engineering, Fifth Edition, Jones and Bartlett Publishers, Boston, 2006

9. Jeffrey, A., Complex Analysis and Application, Second Edition, Chapman and Hall/CRC, New York, 2006


BRA ‐ II MATH 504: ALGE

Course contents:

Computer codes and Number system : Number systems, binary, octal and hexadecimal system. 4 bit, 6 bit and 8 bit BCD codes. Zone decimal and packed decimal formats. Computer arithmetic, errors. Logic, Truth tables: Conjunction, disjunction, negation, propositions and truth tables, tautologies and contradictions, logical equivalence, algebra and propositions, conditional and biconditional statements, logical implication. Algorithms, flowcharts, pseudocode, and programs: Computer programs variables, constants, flowcharts and their language. Loops, initialization counters, accumulators, DO loops pseudocode programs. Boolean algebra, Logic gates: Boolean algebra, duality, basic theorems. Order and Boolean algebra. Boolean expressions, sum of product form. Logic gates, logic circuits, Minimal Boolean expressions. Combinatorial analysis. Graph Theory: Graphs and multi graphs, Degree of a vertex, deterministic and non‐deterministic automata.

Books Recommended:

1. Stanley, I., Grossman, Applied Linear Algebra, Second Edition, Wadsworth Publishing Co., California, 1984.

2. rs Stroud, K. A., Linear Algebra: Theory and Application, Stanley Thornes Publishe

3. aticians. Ltd., 1978. Graham, A., Matrix Theory and Applications for Engineers and Mathem

4. Graham, A., Nonnegative Matrices and Applications for Engineers and Mathematicians.

5. Lipschutz, S., Essential Computer Mathematics, Mc Graw Hill Inc., 1982.

6. Lennox, S. C., Chadwick, M., Computer Mathematics for Applied Scientists, Second Edition,

Heinemann Educational Books Ltd., London, 1985.

7. Garding and Tambour, Algebra and Switching Circuits, Mc Graw Hill 1988.

8. Mendelson, E., Boolean Algebra and Switching Circuits, Mc Graw Hill 1978.

9. Halmon, P. R., Lectures on Boolean Algebra, Van Nostrand, 1963.

10. Rosen, K. H., Discrete Mathematics and its Applications, Fifth Edition, AT and T Laboratories,

New Jersey, Mc Graw Hill, 2001.

MATH 506: NUMERICAL ANALYSIS – II (2 +1)

Numerical Methods for solving Initial Value Problems: Taylor’s Series method, Euler’s and Modified Euler’s methods. The Runge Kutta Methods. Higher order Initial Value problems and Systems of First order Initial Value problems. Multi‐step Methods of Milne and Adams‐Moulton. Numerical Methods for solving Boundary Value Problems: The Shooting Method, The Difference Equation Methods, Rayliegh‐Ritz, Collocation and Galerkin’s methods. Introduction to Finite Element methods. Characteristic Value Problems: Determination of eigenvalues iteratively, Power and the Inverse Power methods, the QR method for finding eigenvalues of a matrix. Numerical Methods for solving Partial Differential Equations: Representation as difference equations, Laplace and Poisson

r coordinates and their solutions. equation in rectangula

Books Recommended:

1. Allen III, M. B. and Isaacson, E. L., Numerical Analysis for Applied Sciences (Pure and Inc. Applied Mathematics A. Willy‐interscience series texts), John Wiley and Sons

N.Y. 1998 2. Jain, M. K., Iyengar, S. R. K. and Jain, R. K., Computational methods for Partial

Differential Equations. Wiley Eastern Limited, New Delhi, 1991. 3. Jain, M. K., Iyengar, S. R. K. and Jain, R. K., Numerical Methods for Scientific and

Engineering Computations. Wiley Eastern Limited, New Delhi, 1991. 4. Atkinson, K. E., An Introduction to Numerical Analysis, John Wiley and Sons, N.Y.,

1989. 5. Hager, W. W., Applied Numerical Linear Algebra, Prentice Hall International Inc.

Toronto, Canada, 1995. 6. Book Chapra, S. C. and Canale, R. P., Numerical Methods for Engineers, Mc Graw Hill

Co. Toronto, 2000. 7. Mathews, J. H., Numerical Methods for Mathematics, Science and Engineering,

Pentice Hall International Inc, N.J., 1984. 8. b. Gerald, C. F. and Patric, O.W., Applied Numerical Analysis, Addison Wesley Pu

9. Com., 1984. King, J. T., Introduction to Numerical Computation, Mc Graw Hill, N. Y., 1984.

10. Vendergraft, J. S., Introduction to Numerical Computation, Academic Press, NewYork, 1983.



13. Griffiths, D. V. and Smith, I. M., Numerical Methods for Engineers, Second Edition, Chapman and Hall/CRC, New York, 2006.

14. ist, Chapra, S. C., Applied Numerical Method, with MATLAB: For Engineers and ScientSecond Edition, Tufts University, McGraw Hill, 2007.

15. Karris, S. T., Numerical Analysis using MATLAB and Excel, Third Edition, Orchant Pub., 2007.

ICABLE DIFFERENTIAL GEOMETRY ‐ II MATH 508: APPL

Course contents:

Bilinear and quadratic forms. Euclidean spaces. Metrics on affine spaces. Parallelism in affine metric spaces. Vector calculus. Isometrics: killing equation and killing fields, Rotation group. Surfaces. Differential geometry on surfaces. Riemannian geometry curvature, Surfaces geometry in terms of exterior forms Levi‐Civita connection. Covariant derivative.

ture. Connection and curva

Books Recommended:

1. Crampin, M. and Pirani, F. A. E., Applicable Differential Geometry, CUP, 1986. Goetz, A., Introduction to Differential Geometry; Addison Wesley, 1970.

3. ntice Hall Inc., 2.

Milman, R. and Parker, G., Elements of Differential Geometry; Pre

4. 1995. 1977. O’ Neill, B., Elementary Differential Geometry; Academic Press,

rwood, 1976. 5. Chorlton, F., Vector and Tensor Methods, Ellis Ho

ODS OF MATHEMATICAL PHYSICS ‐ II MATH 510: METH

Course contents:

Partial differential equations of Mathematical Physics. Method of separation of variable. Boundary value problems relating vibration of strings and membranes, heat conductionand potential theory. Qualitative theory of differential equations. First order equation and models of population growth. Qualitative analysis of second order autonomous systems. Almost linear systems. Examples of damped pendulum and Lotka‐Volterra equations.

thod. Lorenz equations: chaos and strange attractors. Liapunov’s second me

Books Recommended:

1. Boyce, W. E. and De Prima, R. C., Elementary Differential Equations and Boundary Value Problems; Fifth Edition, Wiley, New 1992

2. F nChurchil, R.V. and Brown, J. W., ourier Series a d Boundary Value Problems, Third Edition, Mc Graw Hill Kogakusha, Tokyo1978

3. Finnery, R. L. and Ostberg, D. R., Elementary Differential Equations with Linear Algebra Addison Wesley, Reading, Mass. 1976.

4. Rainville, E. D. and Bedient, P. E., Elementary Differential Equations, Seventh Edition, Macmillan, New York1989

5. CoLeighton, W., First urse in Ordinary Differential Equations, Wadsworth Publishing Co. Belmont, California, 1981.

6. Arrowsmith, D. K. and Place, C. M., Ordinary Differential Equations, Chapman and Hall, 1982.

7. ns, New Barelli, R. L. and Coleman, C. S., Differential Equations, John Wiley and SoYork, 1998.

8. Humi, M. and Miller, W. B., Boundary Value Problem and Partial Differential Equations. PWS‐Kent Publishing Co., Boston, 1992.

9. o Ltd, India, Raisinghania, Ordinary and Partial Differential Equations. Chand S. and C

10. 2007.

11. Kevorkian, J., Partial Differential Equations, 2nd edition, Springer, 1999. Zwillinger D., Handbook of Differential Equations, AK Peters, 1992.

12. Birkhoff, G. and Rota, G. C., Ordinary Differential Equations, Forth Edition, John Wiley and Sons, New York, 1989.


14. rning, O’Neil, P. V., Advanced Engineering Mathematics, Fifth Edition, Cengage Lea2003.

15. Dennis, G Zill, Micheal R Cullen, Differential Equation with Boundary Value Problems, Fifth Edition, Loyala Marymond Uni., Brooks Cole Pub., 2001. Kreyszig, E., Advanced Engineering Mathematics, Ninth Edition, John Willey, 2

17. ol 16. 005.

Pandey R. K., Partial Differential Equation in several complex variable, AnmPublication Pvt. Ltd., 2008.

18. Pandey R. K., Partial Differential Equation in complex variable and integral transforms, Anmol Publication Pvt. Ltd., 2008.

19. O’Neil P. V., Beginning Partial Differential Equations, Second Edition, Wiley Interscience Pub., John Wiley and Sons Inc., 2008.

PU THE S RE MA MATIC

SEVENTH SEMESTER

MATH 601: ABSTRACT ALGEBRA

Course contents:

Group Theory: Groups, Subgroups, cyclic groups, normal subgroups, quotient groups, examples. Homomorphism of groups, the fundamental theorem of homorphism. Isomorphism of groups, the isomorphism theorems. Direct product of groups. Internal and external direct products. Finitely generated Abelian groups. Generators and torsion. The fundamental theorem of F.G. Abelian groups. Applications. Group action on a fixed sets and isotropy subgroups, orbits. Sylow theorems, p‐groups. First, second and third Sylow theorems. Application of the Sylow theory. RING THEORY: Rings. Integral domain. The characteristic of a ring. Fermat’s and group algebra. Quotient rings, ideals, maximal and

prime ideals. Ring homomorphism: Definition, properties, prime fields. Fundamental theorems of homorphism and isomorphism. Polynomial rings, the evaluation modules, ideals, Isomorphism theorem. Near rings, subnear rings, near ring modules, isomorphism theorem.

Books R ecommended:

1. Fraleigh, J. B., A First Course in Abstract Algebra, Third Edition, Addison Wesley Publishing Co., 1982.

2. ra, Allenby, R. B. J. T., Rings, Fields and Groups An Introduction to abstract Algeb

3. Edward Arnold Ltd., 1983. Burnisde, W., Theory of Groups of Finite Order, Second Edition, Dover, N.Y., 1955.

4. Hall, M. Nesbitt, C. J. and Thrall, R. H., Rings with Minimum Conditions, Ann Arbor, Univ. of Michigan Press, 1944.

5. uArtin, E., Nesbitt, C. J. and Thrall R. H., Rings with Minim m Conditions, Ann Arbor, Univ. of Michigan Press, 1944.

6. Mecoy, N. H., Rings and Ideals, Cerus Monograph No.81 Buffalu, The Mathematical Association of America.

7. Stewart, J. N., Galois Theory, Chapman and Hall, London, 1973.

Indiana, 1964. 8. Artin, E., Galois Theory, University of Notre Dam Press,

9. Garling, D. J. H., A Course in Galois Theory, C.U.P., 1986.

10. Adamson, I. T., Introduction to Field Theory, Oliver and Boyd, 1964,

rica, 1963. 11. Albert, A A., Studies in Modern Algebra, Mathematical Association of Ame

12. Gaal, L., Classical Galois Theory with Examples, Markham, Chicago, 1971.

13. Hadlock, C. R., Field Theory and its Classical Problems, Carus Monograph, Mathematical

Association of America, 1978.

AL ANALYSIS ‐ I MATH 611: FUNCTION

Course contents:

Metric spaces. Euclidean and Unitary spaces. Complete, compact and separable metric spaces. Sets of first and second category. Bires category theorem Equicontinuity. Arzolla’s theorem. Normed vector spaces and Banach spaces. Bounded linear transformations and functionals, Dual spaces, Hahn‐Banach theorem, Uniform boundedness. Banach‐Steinhaus theorem. Open Mapping and closed graph theorems and their applications. The dual of normed spaces, Adjoints.

Books Recommended:

1. ons, Kreyszig, E., Introductory functional analysis with applications John Wiley and S1978.

2. Nachbin, L., Introduction to Functional Analysis: Branch Space and Differential Calculus, Marcel Dekker. Inc.1981.

3. ge University Press, Davis, E. B., Spectral Theory and Differential Operators, Cambrid

4. 1995. Limaye, B.V., Functional Analysis, Wiley Eastern Limited, 1980.

5. Devito, C. L., Functional Analysis and Linear Operator Theory, Addison Wesley

6. Publishing CO., 1990.

7Siddiqui, A. H., Functional Analysis with applications, Tata McGraw Hill, 1986.

8. . Vulik, B. Z, Introduction to Functional Analysis, Pergamon, 1963.

w Hill, 1998. 9. 997.

Simmons, G. F., Introduction to Topology and Modern Analysis, McGra

10. Goffman, C. and Pedrick, G., First Course in Functional Analysis, Prentice Hall, 1Taylor, A. E., Introduction to Functional Analysis, Prentice Hall, 1979.

11. Somasundaram, D., A First Course in Functional Analysis, First Edition, Narosa lhi, 2006. Publishing House, New De

URE THEORY ‐ I MATH 605: MEAS

Course contents:

Outer and inner measures. Measurable and non‐measurable sets and functions. Uniform convergence and convergence in measure. Functions of bounded variation. Absolute continuity.

Books Recommended:

1. Halmos, P. R., Measure Theory, Van Mostrand, Springer Pub., 1978.

2. Kestelman, H., Modern theories of Integration, Second Edition, Dover Pub., 1980.

rentice Hall, 1997. 3. Goffman, C. and Pedrick, G., First course in Functional Analysis, P

1988. 4. Halsey Royden, Real Analysis, Third Edition, Prentice Hall,

5. Donald L. Cohn, Measure Theor u n vers ty, 1997. y, S ffolk U i i

EIGHTH SEMES

IS THEORY AND ITS APPLICATIONS.

T ER

MATH 602: GALO

Course contents:

The theory of fields, field extension, algebraic extensions, monomorphism of algebraic extension. Test for irreducibility, Einstein’s criterion. Other methods for establishing irreducibility. Rules and compass construction. Splitting fields, the extension of monomorphism with examples. The algebraic closure of a field. Normal extensions, separability. Galois extensions, differentiation, the Frobenius monomorphism. Inseparable polynomials, automorphisms and fixed fields. The Galois theory, the theorem on natural irrationalities.

Books R ecommended:

1. ebra, Fraleigh, J. B., A First Course in Abstract Alg Third Edition, Addison Wesley Publishing Co., 1982.

2. ra, Allenby, R. B. J. T., Rings, Fields and Groups: An Introduction to abstract Algeb

3. Edward Arnold Ltd., 1983. Burnisde, W., Theory of Groups of Finite Order, Second Edition, Dover, N.Y., 1955.

4. Hall, M., Nesbitt, C. J. and Thrall, R.H., Rings with Minimum Conditions, Ann Arbor, Univ. of Michigan Press, 1944.

5. . e t r . Artin, E , N sbi t, C. J. and Th all R H., Rings with Minimum Conditions, Ann Arbor, Univ. of Michigan Press, 1944.

6. Mecoy, N. H., Rings and Ideals, Cerus Monograph No.81 Buffalu, The Mathematical Association of America.

7. Stewart, J. N., Galois Theory, Chapman and Hall, London, 1973.

Indiana, 1964 8. Artin, E., Galois Theory, University of Notre Dam Press,

9. Garling, D. J. H., A Course in Galois Theory, C.U.P., 1986.

10. Adamson, I. T., Introduction to Field Theory, Oliver and Boyd, 1964,

rica, 1963. 11. Albert, A. A., Studies in Modern Algebra, Mathematical Association of Ame

12. Gaal, L., Classical Galois Theory with Examples, Markham, Chicago, 1971.

13. Hadlock, C. R., Field Theory and its Classical Problems, Carus Monograph, Mathematical

Association of America, 1978.

TIONAL ANALSYSIS ‐ II MATH 612: FUNC

Course contents:

Hilbert spaces. Projection theorem. Orthonormal and complete orthonormal sets. Operators in Hilbert spaces Invariant subspaces and projections. Spectral mapping

theorem in finite dimensional Hilbert spaces. Banach algebras, C‐ and B‐ algebras. Krein‐ azur theorems. Millman and Gelfand‐M

Books Recommended:

1. Sons, Kreyszig, E., Introductory functional analysis with applications John Wiley and1978.



4. 1995. Limaye, B. V., Functional Analysis, Wiley Eastern Limited, 1980.



7Siddiqui, A. H., Functional Analysis with applications, Tata McGraw Hill, 1986.


w Hill, 1998. 9. ice Hall, 1997.

Simmons, G. F., Introduction to Topology and Modern Analysis, McGraGoffman, C. and Pedrick, G., First Course in Functional Analysis, Prent

Functional Analysis, Prentice Hall, 1979. 10. Taylor, A. E., Introduction to

THEORY ‐ II MATH 606: MEASURE

Course contents:

Integrable functions. Convergence and mean convergence. L spaces. Signed measure. rem and derivatives. Haar measure. Raydon‐Nikodym theo

Books Recommended:

1. Halmos, P. R., Measure Theory, Van Mostrand, Springer Pub., 1978.

2. Kestelman, H., Modern theories of Integration, Second Edition, Dover Pub., 1980.

is, Prentice Hall, 1997. 3. Goffman, C. and Pedrick, G., First course in Functional Analys

4. Royden, H., Real Analysis, Third Edition, Prentice Hall, 1988.

5. Cohn, D. L., Measure Theory , Suffolk University, 1997

APP ATH ICS LIED M EMAT

S

TIONAL ANALYSIS ‐ I

EVEN SEMESTER TH

MATH 611: FUNC

Course contents:

Metric spaces. Euclidean and Unitary spaces. Complete, Compact and Separable Metric spaces. Sets of first and second category. Bires Category Theorem. Equicontinuity. Arzolla’s Theorem. Normed vector spaces and Banach spaces. Bounded linear transformations and functionals, Dual spaces, Hahn‐Banach theorem, Uniform boundedness. Banach‐Steinhaus theorem. Open Mapping and closed graph theorems and

dual of normed spaces, Adjoints. their applications. The

Books Recommended:

1. ons, Kreysziz, E., Introductory functional analysis with applications John Wiley and S1978.



4. 1995. Limaye, B. V., Functional Analysis, Wiley Eastern Limited, 1980.



7 Siddiqui, A. H., Functional Analysis with applications, Tata McGraw Hill, 1986.


w Hill, 1998. 9. 997.

Simmons, G. F., Introduction to Topology and Modern Analysis, McGra

10. Goffman, C. and Pedrick, G., First Course in Functional Analysis, Prentice Hall, 1 Taylor, A. E., Introduction to Functional Analysis, Prentice Hall, 1979.

11. Somasundaram, D., A First Course in Functional Analysis, First Edition, Narosa Publishing House, New Delhi, 2006.

EMATICAL STATISTICS – I (2 + 1) MATH 631: MATH

Course contents:

Combinatorics, Axioms of Probabilities, Conditional probabilitiy and independence. Random variables, Distribution function, Discrete probability distributions: Uniform, Bernoulli, Binomial, Poisson, Negative Binomial and Hyper geometric distributions. Continuous probability functions: Uniform, Exponential, Gamma, Beta, Weibull, Cauchy, and Rayleigh Distributions. Expected value of a random variable, Moment generating

enerating function. function, Probability g

Books R :ecommended

1. Robert, V. H., Allen, C. and Joseph, W. M., Introduction to Mathematical Statistics, Fourth Edition, Prentice Hall, 2004.

2. Elliot, A.T. and Robert V. H., A brief Course in Mathematical Statistics, Amazon, Com. Inc.2007.

3. Richard, J. L. and Morris L. M., An introduction to Mathematical Statistics, Prentice Hall, 2000.

4. J. ., MLarsen, R. and Marx, M. L An introduction to athematical Statistics and its Applications, Englewood Cliffs, N. J., 2005.

5. Bain, L. J. and Engelhardt. M., Introduction to probability and Mathematical Statistics, Third Edition, Dluxbury Press, 1992.

6. Balakrishnan, N. and Cohen, A. C., Order Statistics and Inference, Academic Press, Inc., New York, 1991.

7. Casella, G. and Berger. R. L., Statistical Inference, Second Edition, Wadsworth Publishing Company, 1999.

8. , Hoog, R. V. and Tanis E. A. , Probability and Statistical Inference, Seventh Edition

9. Prentice Hall, 2005. David, H. A., Order Statistics, Second Edition, John Wiley and Sons, New York, 1981.

10. Theory of Statistics, Mood, A. M., Graybill, F. A. and Boes D. C., Introduction to the

11. Third Edition, McGraw‐Hill, 1974. Shao, J., Mathematical Statistics, Second Edition, Springer, 2003.

12. Wakerly, D. R., Mendenhall, W. W. and Scheaffer, R. L., Mathematical Statistics with applications, Duxbury Press, 2007.

13. Johnson N. L., Kotz S., and Balakrishan N., Continuous Univariate Distribution, Vol John Wiley and Sons Inc., 2000. 1,Second Edition, Wiley Interscience Pub.,

ATIONS RESEARCH ‐ I (2 + 1) MATH 633: OPER

Course contents:

Introduction: Origins, Nature and Impact of Operation Research. The Operations Research Modeling approach, Linear programming (LP): Introduction and Methods for solving LP models and assumptions in LP Model Theory of the simplex Method, Breaking in the simplex Method, Modified Simplex Methods. Applications of simplex and modified simplex methods, Duality theory, Sensitivity analysis and applications to linear programming, Dual Simplex Method, Parametric Linear Programming. Upper Bound Technique, Extension of Linear Programming to Transportation and Assignment Models and Methods for solving these models, Integer and Nonlinear Programming Methods and their application

Dynamic Programming (DP): Introduction, characteristics of DP, Determinate DP, DP, and their applications. Probabilistic

Labs / Drills:

Linear Programming (Three Cases), Simplex Method (Three Cases), Duality and Sensitivity Analysis (Four Cases), Transportational and Assignment Method (Three Cases), Integer Programming (Four Cases), Non Linear Programming (Three Cases) and Dynamic Programming (One Case).

Books Recommended:

1. Saaty, L. S., Mathematical Methods of Operations Research, John Wiley, 1986. Rao, S. S., Optimization Problem, Willey Eastern, New Delhi, 1987.

3. n Fransisco, 2.

Killier, F. S. and Lieberman, G. J., Operations Research, Holden Day, Sa

4. Calif,. 1988.

5. Mustafi, C. K., Operations Research, Willey Eastern, New Delhi, 1982. Gupta, P. K. and Hira, D. S., Operations Research S. Chand, New Delhi, 1994.

6. and Application, Moder and Elmaghrby, Hand Book of Operation Research Models

7. Vols 1 and 2, Van Nostrand Renhold, 1982.

8. Taha, S. A., Operation Research, Willey Eastern, New Delhi, 1996. Minkash, T. A., The Optimization Problem, Eastern Publishers, 1992.

9. Loomba, N. P., Management – A quantitative Perspective, Barnch College, City University of New York, 1978.

10. Hiller, F., Introduction to Operational Research, Stanford University, Eighth Edition, 2005.

EI

TIONAL ANALSYSIS ‐ II

GHTH SEMESTER

MATH 612: FUNC

Course contents:

Hilbert spaces. Projection theorem. Orthonormal and complete orthonormal sets. Operators in Hilbert spaces Invariant subspaces and projections. Spectral mapping theorem in finite dimensional Hilbert spaces. Banach algebras, C‐ and B‐ algebras. Krein‐

azur theorems. Millman and Gelfand‐M

Books Recommended:

1. Kreyszig, E., Introductory functional analysis with applications, John Wiley andSons, 1978.

2. Nachbin, L., Introduction to Functional Analysis: Branch Space and Differential Calculus, Marcel Dekker. Inc., 1981.

3. ge University Press, Davis, E.B., Spectral Theory and Differential Operators, Cambrid

4. 1995. Limaye, B.V., Functional Analysis, Wiley Eastern Limited, 1980.



7Siddiqui, A. H., Functional Analysis with Applications, Tata McGraw Hill, 1986.

8. . Vulik, B. Z., Introduction to Functional Analysis, Pergamon, 1963.

w Hill, 1998. 9. ice Hall, 1997.

Simmons, G. F., Introduction to Topology and Modern Analysis, McGraGoffman, C. and Pedrick, G., First Course in Functional Analysis, Prent

, Prentice Hall, 1979. 10. Taylor, A. E., Introduction to Functional Analysis

MATH 632: MATHEMATICAL STATISTICS – II (2 + 1)

Course contents:

Distribution of two random variable, the correlation coefficient, conditional distributions, conditional expectations, transformation of random variables, chi‐square, T and F distributions. Point estimation, sufficiency and completeness, confidence intervals for means, difference of two means, variance and proportions, sample size, order statistics, Asymptotic distribution of maximum likelihood estimators.

Test of statistical hypotheses: testing mean with known / unknown variance, equality of two normal distributions, chi‐square goodness of fit test, contingency tables, Kolmogrov‐Smirnov goodness of fit test,

Linear and Multiple regressions. Power of a statistical test, best critical regions, likelihood ratio test

Books R :ecommended

1. Robert, V. H., Allen, C. and Joseph W. M., Introduction to Mathematical Statistics, Prentice Hall, 2004.

2. Elliot, A.T. and Robert V. H., A brief Course in Mathematical Statistics, Amazon, Com. Inc.2007.

3. , L iRichard J. . and Morris L. M., An ntroduction to Mathematical Statistics, Prentice Hall, 2000.

4. J. ., MLarsen, R. and Marx, M. L An introduction to athematical Statistics and its Applications, Englewood Cliffs, N. J., 2005.

5. Bain. L. J. and Engelhardt. M., Introduction to probability and Mathematical Statistics, Third Edition, Dluxbury Press, 1992.

6. Balakrishnan, N. and Cohen, A. C. Order Statistics and Inference, Academic Press, Inc., New York, 1991.

7. LCasella, G. and Berger, R. . Statistical Inference, Second Edition, Wadsworth Publishing Company, 1999.

8. , Hoog, R. V. and Tanis, E. A. Probability and Statistical Inference, Seventh Edition

9. Prentice Hall, 2005. David, H. A., Order Statistics, Second Edition, John Wiley and Sons, New York, 1981.

10. Theory of Statistics, Mood, A. M., Graybill, F. A. and Boes D. C., Introduction to the

11. Third Edition, McGraw‐Hill, 1974. Shao, J., Mathematical Statistics, Second Edition Springer, 2003.

12. Wakerly, D. R., Mendenhall, W. W. and Scheaffer, R. L., Mathematical Statistics with applications, Duxbury Press, 2007.

13. Johnson N. L., Kotz S. and Balakrishan N., Continuous Univariate Distribution, Vol 1, hn Wiley and Sons Inc., 2000. Second Edition, Wiley Interscience Pub., Jo

ATIONS RESEARCH – II (2 + 1) MATH 634: OPER

Course contents:

Stochastic process, Markov Chains, Chapman‐ Kolmogrov Equation, Classification of Markov Chains, long‐run Properties, first Passage time, Absorbing states, Continuous time Markov chain.

Inventory models, machine interference problem forecasting techniques. Markovian process. Decision analysis, reliability theory. Renewal process. Queuing theory, ams and graph theory. CPM and PERT theories. Inventory models. Related software application models. Simulations, generation of random numbers. Models for bond analysis yield the

ation immunization and convexity. maturity, dur

Labs / Drills:

Network Optimization (Three Cases), Decision Analysis (Two Cases), Queuing Theory (Two Cases),

r Cases). Inventory Theory (Fou

Books Recommended:

1. Saaty, L. S., Mathematical Methods of Operations Research, John Wiley, 1986. Rao, S. S., Optimization Problem, Willey Eastern, New Delhi, 1987.

3. n Fransisco, 2.

Killier, F. S. and Lieberman, G. J., Operations Research, Holden Day, Sa

4. Calif,. 1988.

5. Mustafi, C. K., Operations Research, Willey Eastern, New Delhi, 1982. Gupta, P. K. and Hira, D. S., Operations Research S. Chand, New Delhi, 1994.

6. and Application, Moder and Elmaghrby, Hand Book of Operation Research Models

7. Vols 1 and 2, Van Nostrand Renhold, 1982.

8. Taha, S. A., Operation Research, Willey Eastern, New Delhi, 1996. Minkash, T. A., the Optimization Problem, Eastern Publishers, 1992.

9. Loomba, N. P., Management – A quantitative Perspective, Barnch College, City University of New York, 1978.

10. Hiller, F., Introduction to Operational Research, Stanford University, Eighth Edition, 2005

Co ntents of Optional Courses For B.S. Program inmatics For Seventh & Eighth Semesters Mathe

UTATIVE RINGS. MATH 603: COMM

Course contents:

Ideals and their operations. Maximal prime and primary ideals. Certain radicals of a ring. Rings with chain conditions. Divisibility theory in integral domains. Rings of fractions.

integral domains. Polynomial rings and

Books Recommended:

1. Herstien, I. H., Topics in Algebra, Second Edition, John Wiely, 1975

.

.

2. Arfine, E., Galois Theory, Second Edition, Notre Dame Press, 1966

3. Keplansky, I., Fields and Rings, University of Chicago Press, 1969.

4. Atiya, M. F. and Macdonald, I. J., Introduction to Commutative Algebra, Addison Wesley, 1969.

tive Rings, University of Chicago press, 1974. 5. Kaplansky, I. H., Commuta

THEORY. MATH 604: FIELD

Course contents:

Field extensions, the degree of an extension. Galois group. Normality and separability. Field degrees and group orders Normal closures. The Galois correspondences.

by radicals. Finite fields. Solution of equations

Books Recommended:

. 1. Herstien, I. H., Topics in Algebra, Second Edition, John Wiely, 1975

. 2. Arfine, E., Galois Theory, Second Edition, Notre Dame Press, 1966

3. Keplansky, I., Fields and Rings, University of Chicago Press, 1969.

4. Atiya, M. F. and Macdonald, I. J., Introduction to Commutative Algebra, Addison Wesley, 1969.

, university of Chicago press, 1974. 5. Kaplansky, I. H., Commutative Rings

ABILITY THEORY ‐ I MATH 607: SUMM

Course contents:

Metric transformations. Toeplitz steinhaus and Kojima Shur theorems and their integral andlogues. Regularity, consistency, equivalence and inclusions of some methods of summability.

Books Recommended:

1. Hardy, G. H., Divergent Series, Clarendon Press, 1976. Peterson, C. M., Regular Matrix Transformations, McGraw Hill,1980.

3. Knopp, K., Theory and Applications of Infinite Series, Blakie and sons, 1987. 2.

MATH 608: SUMMABILITY THEORY ‐ II

Course contents:

Holder and Cesaro’s means. Mercer’s theorem. Summability of integrals. Euler, Borel s of summability and Hausdorff method

Books Recommended:

1. Hardy, G. H., Divergent Series, Clarendon Press, 1976. Peterson, C. M., Regular Matrix Transformations, McGraw Hill, 1980.

ations of Infinite Series, Blakie and sons, 1987. 2. 3. Knopp, K., Theory and Applic

BRIC TOPOLOGY ‐ I MATH 609: ALGE

Course contents:

Categories and functors. Homotopy, fundamental group and covering spaces. nd approximations. Simplical complexes a

Books Recommended:

1. Spanier, E. H., Algebraic Topology, McGraw Hill, 1982. to Homology Theory, C.U.P., 1987. 2. Hilton, P. J. and Wiley, Introduction

BRIC TOPOLOGY – II MATH 610: ALGE

Course contents:

Homology theory of simplical and chain complexes. Singular homology theory, Exact Homotopy sequences.

Books Recommended:

1. Spanier, E. H., Algebraic Topology, McGraw Hill, 1982. Homology Theory, C.U.P., 1987. 2. Hilton, P. J. and Wiley, Introduction to

E THEORY – I (2 + 1) MATH 613: SPLIN

Course contents:

Affine Maps; Translation, Rotation, Reflection, Stretching, Scaling and Shear. Barycentric combination. Convex combination. Convex Hull. Forms of Parametric curve: Algebric form, Hermit form, Control point form, Bernstein Bezier form and their matrix forms. Algorithm to compute Bernstein Bezir form. Properties of Bernstein Bezier form, Convex Hull property. Affine invariance property, Variation diminishing property. Rational

quadratic form. Rotational cubic form. Tensor product surface. Natural spline. Cardinal spline. Perodic spline on uniform mesh. Representation of spline and it’s different forms. Natural spline and periodic spline in terms of polynomials and power truncated functions. Odd degree spline. Existence theorem. Existence and uniqueness of natural and periodic

rem. spline. Remainder theo

Books Recommended:

1. Schumaker L. L., Spline Functions: Basic Theory, John Wiley, Third Edition, Cambridge University Press, 2007.

2. Lai, M. J. and Schumaker L. L., Spline Functions on Triangulations(Encyclopedia of Mathematics and its Applications).First Edition, Cambridge University Press, 2007.

3. Bartels, R. H., Bealty J. C. and Beatty J. C., An Introduction to Spline for use in Computer Graphics and Geometric Modeling, Morgan Kaufmann Publisher, 2006.

4. Wang, R. H., Multivariate Spline Functions and Their Application, Mathematics and it’s Applications, Science Press, Kluwer Academic Publishers, 2005.

5. Farin, G., Curve and Surfaces for Computer Aided Geometric Design: A practical Guide, Academic Press Inc., 2002.

6. Kouncher, O., Multivariate Polyspline: Applications to numerical and Wavelet Analysis, SA, 2001. ,First Edition, Academic Press, Harcourt Science and Technology, U

. 7. DeBoor, C., A Practical Guide to Spline, Springer Verlag, 2001

8. Brannan, D. A., Geometry, Cambridge University Press, 1999.

9. Knoth, G. G., Interpolating Cubic Spline (Progress in Computer Science and Applied 1999.Logic (PCS)), First Edition, Birkhauser Boston,

10. Enbank, R. L., Nonparametric Regression and spline smoothing, Second Edition, CRC, 1999

11. Spath, H., Two Dimensional Spline Interpolating Algorithms, AK Peters Ltd., 1995.

polating Algorithms, AK Peters Ltd., 1995. 12. Spath, H., One Dimensional Spline Inter

E THEORY – II (2 + 1) MATH 614: SPLIN

Course contents:

Interpolatory cubic splines. The representation of s in terms of the values Mi = s(2)(xi), i =0,1,2…..,k.

The representation of s in terms of the values mi = s(1)(xi), i = 0,1,2,……k. Quadratic Hermit spline. Theorems regarding error analysis. Theorems regarding to convergence of the D1, D2, Natural and periodic spline. End conditions for cubic Hermit spline interpolation. E(α) –

ensional Spline and Multivariate Spline also. cubic splines. Two Dim

Books Recommended:

1. Schumaker L. L., Spline Functions: Basic Theory, John Wiley, Third Edition, Cambridge University Press, 2007.

2. Lai, M. J. and Schumaker L. L., Spline Functions on Triangulations(Encyclopedia of Mathematics and its Applications).First Edition, Cambridge University Press, 2007.

3. Bartels, R. H., Bealty J. C. and Beatty J. C., An Introduction to Spline for use in Computer Graphics and Geometric Modeling, Morgan Kaufmann Publisher, 2006.

4. Wang, R. H., Multivariate Spline Functions and Their Application, Mathematics and it’s Applications, Science Press, Kluwer Academic Publishers, 2005.

5. Farin, G., Curve and Surfaces for Computer Aided Geometric Design: A practical Guide, Academic Press Inc., 2002.

6. Kouncher, O., Multivariate Polyspline: Applications to numerical and Wavelet Analysis, SA, 2001. ,First Edition, Academic Press, Harcourt Science and Technology, U

. 7. DeBoor, C., A Practical Guide to Spline, Springer Verlag, 2001

8. Brannan, D. A., Geometry, Cambridge University Press, 1999.

9. Knoth, G. G., Interpolating Cubic Spline (Progress in Computer Science and Applied Logic (PCS)), First Edition, Birkhauser Boston, 1999.

10. Enbank R. L., Nonparametric Regression and spline smoothing, Second Edition, CRC, 1999

11. Spath, H., Two Dimensional Spline Interpolating Algorithms, AK Peters Ltd., 1995.

ine Interpolating Algorithms, AK Peters Ltd., 1995. 12. Spath, H., One Dimensional Spl

LEX ANALYSIS ‐ I MATH 615: COMP

Course contents:

Zeros and poles. Poisson‐Jensen formula. Analytic continuation. Lindeloff extension. Integral functions. Picard’s theorem.

Books Recommended:

1. Boas, R. P., Entire Functions, Academic Press, 1989

3.

. 2. Hayman, W. K., Meromorphic Functions, C.U.P., 1960.

Titohmarsh, E. C., Theory of Function, C.U.P., 1987. 4. , Markushevick, A. I., Theory of Functions of a Complex Variable, Vol. II, Prentice Hall

1965. 5. Sharma, S. C., Complex Integration, First Edition, Discovery Publishing House, New

Delhi, 2007.

LEX VARIABLE ‐ II MATH 616: COMP

Course contents:

Meromorphic functions. Navanlinnas fundamental theorems. Deficient values and and exceptional theorems of Polya. functions, derivatives

Books Recommended:

1. Boas, R. P., Entire Functions, Academic Press, 1989

3.

. 2. Hayman, W. K., Meromorphic Functions, C.U.P., 1960.

Titohmarsh, E. C., Theory of Function, C.U.P., 1987. 4. ce Hall, Markushevick, A. I., Theory of Functions of a Complex Variable, Vol. II, Prenti

1965. 5. Jeffrey, A., Complex Analysis and Application, Second Edition, Chapman and

Hall/CRC, New York, 2006. 6. Sharma, S. C., Complex Integration, First Edition, Discovery Publishing House, New

Delhi, 2007.

VE GEOMETRY ‐ I MATH 617: PROJECTI

Course contents:

Projective properties of conics, Chasle’s theorem, projective generation of the conic. Homographic correspondences on the conics. Pascal’s theorem. Linear systems of conics.

geometry. Relation to Euclidean

Books Recommended:

1. Maxwell, E. A., Methods of Plane Projective Geometry, C.U.P.,1993 2. nsions. Maxwell, E. A., General Homogenous CO‐ordinates in Space of Three Dime

C.U.U.,1999 gebraic Projective Geometry, C.U.P.,1987 3. Sample, J. G. and Kenton, G. T., Al

MATH 618: PROJECTIVE GEOMETRY ‐ II

Course contents:

Projective geometry of three dimensions, point, straight line and plane. Duality, cross s. Line geometry. Twisted cubic. ratios, Quadric surface

Books Recommended:

1. Maxwell, E. A., Methods of Plane Projective Geometry, C.U.P.,1993 2. sions. Maxwell, E. A., General Homogenous CO‐ordinates in Space of Three Dimen

C.U.U.,1999 Algebraic Projective Geometry, C.U.P.,1987 3. Sample, J. G. and Kenton, G. T. ,

SICAL MECHANICS ‐ I MATH 619: CLAS

Course contents:

Calculus of variations. Derivation of Euler’s equations and their solutions in special cases. Generalized coordinates. Lagrange’s and applications. Kinematics of a rigid body. Rigid body equations of motion. Euler’s equations. Motion of a heavy symmetrical top.

. Relativistic mechanics

Books Recommended:

1. Sheck, F., Mechanics, Springer Verlay, Berlin, 1988.

3. 2. Goldstein, H., Classical mechanics, Addision Wesley, 1962.

Meirovitch, L., Methods of Analytical Dynamics, McGraw Hill 1970. 4. ic Marion, J. B., Classical Dynamics of Particles and Systems, Second Edition, Academ

5. Press, 1970. Corben, H. C. and Stehle, P., Classical Dynamics, Second Edition, John Wiley, 1960.

6. d, Rund, H., The Hamilton Jacobi Theory in the Calculus of Variations, D. Van Nostran1966.

7. ons of First Caratheodory, C., Calculus of Variations and Partial Differential/Equati

8. Order, Part I, Holden Day, 1965. Taylor, E. F. and Wheller, J. A., Spacetime Physics, W.H. Freeman, 1966.

9. Meirovitch L., Methods of Analytical Dynamics, First edition, McGraw Hill, New York, 2007

SICAL MECHANICS ‐ II MATH 620: CLAS

Course contents:

Hamilton’s equations. Least action principle. Hamilton’s equations. Contact transformations. Symplectic structure on the phase space. Hamilton Jacobi equations. Small

ation to continuous systems and fields. oscillations. Generaliz

Books Recommended:

1. Sheck, F., Mechanics, Springer Verlay, Berlin, 1988.

3. 2. Goldstein, H., Classical mechanics, Addision Wesley, 1962.

Meirovitch, L., Methods of Analytical Dynamics, McGraw Hill,1970. 4. ic Marion, J. B., Classical Dynamics of Particles and Systems, Second Edition, Academ

5. Press, 1970. Corben, H. C. and Stehle, P., Classical Dynamics, Second Edition, John Wiley, 1960.

6. d, Rund, H., The Hamilton Jacobi Theory in the Calculus of Variations, D. Van Nostran1966.

7. ons of First Caratheodory, C., Calculus of Variations and Partial Differential/Equati

8. Order, Part I, Holden Day, 1965. Taylor, E. F. and Wheller, J. A., Spacetime Physics, W.H. Freeman, 1966.

9. Meirovitch L., Methods of Analytical Dynamics, First edition, McGraw Hill, New York, 2007

MATH 621: FLUID DYNAMICS – I (2 + 1)

sics in B.A. / B.Sc. / B.S. Prerequisite: Phy

Course contents:

General introduction. Fluid Properties: Density, Specific Volume, Specific gravity, Pressure, Viscosity, temperature, Thermal Conductivity, Vapour Pressure, Bulk modulus of Elasticity. Kinematics of the flow field: Description of fluid motion, Lagrangian and Eulerian methods, Steady and Unsteady Flow, Uniform and Non‐uniform flows, Line of flows, Streamlines, Stream Surfaces and Stream tube, streak lines, Substantial or Material Derivative, The Reynolds Transport theorem. Differential form of conservation Equations (Continuity, Navier‐ Stokes Equation (NSE), and Energy Equation), Vorticity, rotational and irrotational motions, Existence of Streamfunction, Potential flows (uniform, sourse, sink, vortex, forced vertex, free vortex, combinational vortex, Doublet, source in a uniform stream (half body)) and spiral. Bernoulli Equation, Circulation. Exact Solutions of the Navier Stokes equations: Planes, Couette Flow, Generalized plane Couette flow, plane poiseulle flow, flow between co‐axial and circular pipes/cylinder, Impulsive and oscillatory motion of an infinite flat plate, pulsatile flow between parallel surfaces. Dimensions, Dimensional Homogeneticity, Dimensionless Parameters, Dimensional analysis and

mic similitude. Dyna

Labs:

1. available software(s) such as Plotting of implicit streamfunctions using

2. Mathematica/ MATLAB/ MAPLE, etc. Computing and plotting of Exact solution of NSE.

3. To study problems in Dimensional Analysis and Dynamic similitude with the aid of software(s).

Books Recommended:

1. echanics, Fifth Munson, B. R., Young, D. F and Okiishi, T. H., Fundamentals of Fluid M

2. Edition, John Wiley Sons, N. Y., 2005. Panton, R. L., Incompressible Flows, John Wiley and Sons, N.Y., 2005.

3. Batchelor, G.K., An Introduction to Fluid Dynamics, Cambridge University Press, 2008.

4. pplications, Cengel, Y. A. and Cimbala, J. M., Fluid Mechanics: Fundamentals and A

5. McGraw‐ Hill, Higher Education, 2008. Thompson, P. A., Compressible Fluid Dynamics, McGraw ‐ Hill, 1972

6. O’ Neill, M. E. and Cholton, F., Ideals and Incompressible Fluid Dynamics, Ellis Horwood Ltd, West Sussex, England, 1986.

7. elhi, Bansal, J. H., Viscous Fluid Dynamics, Oxford and IBH Publishers Co, New D

8. 2000. Acheson, D. J., Elementary Fluid Dynamics, Clarendon Press, Oxford, 1990.

9. s, Kuethe, A. M. and Chow, C. Y., Foundation of Aerodynamics, John Wiley and Son

10. N.Y., 1986. Shivamaggi, K. B., Theoretical Fluid Dynamics, Princeton Hall, New Dehli, 1998.

11. ill Cengel, Y. A., Thermodynamics An Engineering Approach, Fifth Edition, McGraw HHigher Education, 2006.

12. Crowe, C. T., Elger, D. F. and Roberson J. R., Engineering Fluid Mechanics, Seventh Edition, John Wiley and Sons, Inc, 2001.

13. Finnemore, E. J., and Franzini, J. B., Fluid Mechanics with Engineering Applications,Tenth Edition, McGraw Hill, New York, 2002.

. Third Edition, McGraw Hill, New York, 2007. 14. Cengel, Y. A., Heat and Mass Transfer

DYNAMICS – II (2 + 1) MATH 622: FLUID

Course contents:

Integral form of conservational equations (mass, momentum and energy) and applications. Open Channel flow: General characteristics of Open Channel flow, surface waves, specific energy, channel depth variation, the Chezy and Manning equations and their applications. Gradually Varied flow, rapidly varied flow, the hydraulic Jump, Sharp‐Crested wires, Broad‐ crested weirs, Underflow Gates. Compressible flow: ideal gases and ideal gas relationships, Mach number, Speed of Sound, categories of Compressible flow, isentropic

f an ideal gas through converging, diverging and converging‐diverging ducts. flow o

Labs:

Designing of open channels(at least three) and ducts (at least two) using available software(s)

Books Recommended:

1. Munson, B. R., Young, D. F and Okiishi, T. H., Fundamentals of Fluid Mechanics, Fifth Edition, John Wiley Sons, N. Y., 2005.

2. Panton, R. L., Incompressible Flows, John Wiley and Sons, N.Y., 2005.

008. 3. Batchelor, G. K., An Introduction to Fluid Dynamics, Cambridge University Press, 2

4. Cengel, Y. A. and Cimbala, J. M., Fluid Mechanics: Fundamentals and Applications, McGraw‐ Hill, Higher Education, 2008.

5. Thompson, P. A., Compressible Fluid Dynamics, McGraw ‐ Hill, 1972

6. O’ Neill, M. E. and Cholton, F., Ideals and Incompressible Fluid Dynamics, Ellis Horwood Ltd, West Sussex, England, 1986.

elhi, 2000. 7. Bansal, J. H., Viscous Fluid Dynamics, Oxford and IBH Publishers Co, New D

8. Acheson, D. J., Elementary Fluid Dynamics, Clarendon Press, Oxford, 1990.

9. Kuethe, A. M. and Chow, C.Y., Foundation of Aerodynamics, John Wiley and Sons, N.Y., 1986.

10. Shivamaggi, K. B., Theoretical Fluid Dynamics, Princeton Hall, New Dehli, 1998.

11. Cengel, Y. A., Thermodynamics An Engineering Approach, Fifth Edition, McGraw Hill Higher Education, 2006.

12. Crowe, C. T., Elger, D. F. and Roberson J. R., Engineering Fluid Mechanics, Seventh Edition, John Wiley and Sons, Inc, 2001.

13. Finnemore, E. J., and Franzini, J. B., Fluid Mechanics with Engineering Applications, Tenth Edition, McGraw Hill, New York, 2002.

sfer. Third Edition, McGraw Hill, New York, 2007. 14. Cengel, Y. A., Heat and Mass Tran

MATH 623: ELECTROMAGNETICS – I

ysics in B. A. / B. Sc. / B. S. Pre Requisite: Ph

Course contents:

Coulomb’s law, Electrostatic Field and Potential, Gauss Law. Energy and Force in Electrostatic Field, Dielectric, Method of Images, Electric dipole, Electric Quadrupole, Electric Octopole, Dipole Radiations, Radiated Energy, Magnetic Dipole Radiations and

Electrostatic Problems Method of Solving the

Books Recommended:

1. Coulson, C. A., Electricity, Fifth Edition, Oliver and Boyd, 1965.

2. Lorrain. P and Crson, D. R., Introduction to Electromagnetic Fields and Waves, Second Edition, W.H. Freeman, 1970.

3. lectricity and Magnetism, Chambers, L. G., An Introduction to the Mathematics of E

4. Chapmean Hall, 1973.

5. Ferraro, V. C. A., Electromagnetic Theory, Athlone, 1967.

6. iley, 1964. Jones, D. S., The Theory of Electromagnetism, Macmillan, 1964.

ields, John W7.

Cheaton, W. B., Elementary Theory of Electric and Magnetic F

8. Cook, D. M., The Theory of the Electromagnetic Field, Prentice Hall, 1975.

9. Shadowitz, A., The Electromagnetic Field, McGraw Hill, 1975. Jackson, J. D., Classical Electrodynamics, Second Edition, John Wiley, 1975.

10. Panofsky, W. K. H. and Phillips, M., Classical Electricity and Magnetism, SecondEdition, Addison Wesley, 1977.

11. Nathan, I., Engineering Electromagnetic, Second Edition, University of Akron, 2004 Springer‐Verlag New York, I.I.C,

TROMAGNMETICS ‐ II MATH 624: ELEC

Course contents:

Steady currents. Magnetic fields of currents, vector potential, magnetic materials and permanent magnetism, Electromagnetic induction. Electromagnetic Waves, Plane Electromagnetic Waves, Plane Harmonic Waves, Waves in Conducting Media, Telegraphy Waves, Inhomogeneous Waves Equations, Poynting Vectors, Scattering Theory, Scattering

nd Sphere. by Circular Cylinder, a

Books Recommended:

1. Coulson, C. A., Electricity, Fifth Edition, Oliver and Boyd, 1965. 2. Lorain, P. and Crson, D. R., Introduction to Electromagnetic Fields and Waves,

Second Edition, W.H. Freeman, 1970. 3. lectricity and Magnetism, Chambers, L. G., An Introduction to the Mathematics of E

4. Chapmean Hall, 1973.

5. Ferraro, V. C. A., Electromagnetic Theory, Athlone, 1967.

6. iley, 1964. Jones, D. S., The Theory of Electromagnetism, Macmillan, 1964.

ields, John W7.

Cheaton, W. B., Elementary Theory of Electric and Magnetic F

8. Cook, D. M., The Theory of the Electromagnetic Field, Prentice Hall, 1975.

9. Shadowitz, A., The Electromagnetic Field, McGraw Hill, 1975. Jackson, J. D., Classical Electrodynamics, Second Edition, John Wiley, 1975.

10. Panofsky, W. K. H. and Phillips, M., Classical Electricity and Magnetism, Second Edition, Addison Wesley, 1977.

11. Nathan Ida, Engineering Electromagnetic, Second Edition, University of Akron, , 2004. Springer‐Verlag New York, I.I.C

TUM MECHANICS – I MATH 625: QUAN

Course contents:

Diract’s bra and ket vectors, Observables, representation theory. Quantum conditions, quantum dynamics. Symmetry properties and conservation theorems. Schrodinger’s and Heisenberg’s pictures. Schrodinger momentum and energy representation. Schrodinger equation, motion in one dimension, Simple harmonic

ave packets. Piecewise continuous potentials. oscillator. Motion of w

Books Recommended:

1. Dirac, P. A. M., The Principles of Quantum Mechanics, Clarendon, 1958. 2. on‐relativistic Theory, Landu, L. D. and Lifshitz, E. M., Quantum Mechanics – N

3. Pergamon, 1959.

4. Merzbacher, E., Quantum Mechanics John Wiley, 1970. Schiffs, L. I., Quantum Mechanics, Third Edition, McGraw Hill, 1979.

5. ley, Dicke, R. H. and Wittke, J.P., Introduction to Quantum Mechanics, Addison Wes

6. 983. 1978.

7. Messiah, A., Quantum Mechanics, Vols. I and II, North Holland, 1961 and 1

8. 70 Mand, M. A., Quantum Mechanics, Butterworths, 1957.

in, 199. , 1966.

Levine, I. N., Quantum Chemistry, Vols. I and II, Allyn and BenjamAnderson, J. M., Mathematics for Quantum Chemistry, Benjamin

y of Radiation, Clarendor, 1960. 10. Histler, W., The Quantum Theor

TUM MECHANICS – II MATH 626: QUAN

Course contents:

Angular momentum. Motion in centrally symmetric field. Hydrogen atom Collision methods. Identical particles and spin. theory. Approximation

Books Recommended:

1. Dirac, P. A. M., The Principles of Quantum Mechanics, Clarendon, 1958. 2. on‐relativistic Theory, Landu, L. D. and Lifshitz, E. M., Quantum Mechanics – N

3. Pergamon, 1959.

4. Merzbacher, E., Quantum Mechanics John Wiley, 1970. Schiffs, L. I., Quantum Mechanics, Third Edition, McGraw Hill, 1979.

5. ley, Dicke, R. H. and Wittke, J.P., Introduction to Quantum Mechanics, Addison Wes

6. 983. 1978.

7. Messiah, A., Quantum Mechanics, Vols. I and II, North Holland, 1961 and 1

8. 70 Mand, M. A., Quantum Mechanics, Butterworths, 1957.

in, 199. , 1966.

Levine, I. N., Quantum Chemistry, Vols. I and II, Allyn and BenjamAnderson, J. M., Mathematics for Quantum Chemistry, Benjamin

uantum Theory of Radiation, Clarendor, 1960. 10. Histler, W., The Q

ivity I MATH 627: Relat

Course contents:

Section A: Contribution of muslims towards conceptual development of principles of relativity (Sadruddin Shirazi, Ibn‐é‐Sina, Naseeuruddin Tusi), contibution of Vigot and Lorentz towards mathematical formulation of relativity, role of Einstein in combining theoretical framework with mathematical formulation, transformation theory (canonical, gauge and coördinate transformations), review of coördinate transformations, homogeneous, isotropic and anisotropic systems and their mathematical description in terms of scalars, vectors and tensors, etc., principle of general covariance, postulates of relativity, weak, medium strong and strong principles of equivalence

Section B: Lorentz transformations (including rotations and velocity in arbitrary direction) and consequences, constancy of velocity of light in free space derived from covariance of Maxwell equations, Poincaré transformations

Section C: Review of modern differential geometry, curvature tensor, Ricci tensor, Bianchi , Weyl tenidentity sor

Section D: Geodesics, geodesic deviation, calculation of geodesic indicating direction of Qibla (Makka)

Section E: Acceleration in terms of curvature tensor, Einstein field equation, Schwarzchild of general relativity solutions, predictions

Books Recommended:

1. Golab, S., Tensor Calculus, North Holland, Amsterdam, 1974. 2. Lawden, D. F., An Introduction to Tensor Calculus, Relativity and Cosmology, John

3. 976. Wiley, New York, 1982.

4. Patharia, R. K., The Theory of Relativity, Second Edition, Pergamon, London, 1Synge, J. L., Relativity: the Special Theory, North Holland, Amsterdam, 1976.

vity: the General Theory, North Holland, Amsterdam, 1980. 5. Synge, J. L., Relati

ivity II MATH 628: Relat

Course contents:

Section A: Solutions of Einstein field equations other than Swarzchild solutions

Section B: Lnon‐relativity black holes, maximal extension and conformal compactification, charged black holes, rotating black holes

ology Section C: Linearized theory of gravity, cosmography, newtionian cosm

, cosmological principle, relativistic cosmology Section D: Hubble’s law

Books Recommended:

1. Golab, S., Tensor Calculus, North Holland, Amsterdam, 1974. 2. Lawden, D. F., An Introduction to Tensor Calculus, Relativity and Cosmology, John

3. 976. Wiley, New York, 1982.

4. Patharia, R. K., The Theory of Relativity, Second Edition, Pergamon, London, 1Synge, J. L., Relativity: the Special Theory, North Holland, Amsterdam, 1976.

ity: the General Theory, North Holland, Amsterdam, 1980. 5. Synge, J. L., Relativ

nomy I MATH 629: Astro

Course contents:

Section A: Orientation of earth, latitude, longitude, meridian, dateline international, poles, Greenwich mean time (GMT), time zones, rotation of earth about its axis, formation of day and night with demonstration, revolution of earth round the sun, tilting of earth axis, seasons, solar calendar, lunar calendar, core of earth, origin of earth magnetism, geographic and magnetic north poles, solar and lunar eclipses, solar system

Section B: Problem‐solving techniques in astronomy, mathematics of instruments used in astronomical observations, errors in astronomical observations

Section C: Gravitational mass and inertial mass, weak principle of equivalence, mass and f o m dweight, act rs in o eling of ‘g’, expression for ‘g’ (inside and outside earth)

Section D: Parts of rocket, rocket and aircraft engines, astrodynamical terminologies, convention to label axes, coördinate transformations, combination of rotations, Euler angles, review of cylindrical and spherical‐polar coördinates, infinitesimal transformations,

ed in astronomy. coördinate systems us

Books Recommended:

1. Baker, R. H., Astronomy, Van Nostrand, Amsterdam, 1998. 2. Battin, R. H., An Introduction to the Mathematics and the Methods of Astrodynamics,

AIAA Education Series, New York, 1987 and 1999. 3. Deusch, R., Orbital Dynamics of Space Vehicles, Prentice Hall, Englewood Cliffs, New

Jersey, USA, 1963. 4. dge, Smart, W. M., Textbook on Spherical Astronomy, Cambridge Univ. Press, Cambri

UK, 1962. physics and Stellar Astronomy, John Wiley, New York, 2001. 5. Swihart, T. L., Astro

nomy II MATH 630: Astro

Course contents:

Section A: Projectile dynamics, orbital and escape velocities, geostationary and polar satellites, satellite‐launch vehicle (SLV), satellite and SLV orbits; down‐range and cross‐

range error for short‐range projectiles; mathematics of inertial‐navigation and telemetry systems

Section B: Review of lagrangian and Hamiltonian dynamics; two‐body problem in plane‐polar‐ and elliptic‐astrodynamical‐coördinate meshes (first one done in detail, second one only introduced)

Section C: Hohmann‐transfer orbit; introduction of control laws (cross‐product, extended‐cross‐product, normal‐component‐cross‐pro‐duct, dot‐product, normal‐component‐dot‐product and ellipse‐orientation steering)

Section D: Introduction of guidance schemes (delta, Lambert and inverse‐Lambert, Q, inverse‐Q and multi‐stage‐Q)

problem and stability of satellites. Section E: Three‐body

Books Recommended:

1. Baker, R. H., Astronomy, Van Nostrand, Amsterdam, 1998. 2. Battin, R. H., An Introduction to the Mathematics and the Methods of Astrodynamics,

AIAA Education Series, New York, 1987 and 1999. 3. Deusch, R., Orbital Dynamics of Space Vehicles, Prentice Hall, Englewood Cliffs, New

Jersey, USA, 1963. 4. dge, Smart, W. M., Textbook on Spherical Astronomy, Cambridge Univ. Press, Cambri

UK, 1962. 5. Swihart, T. L., Astrophysics and Stellar Astronomy, John Wiley, New York, 2001.

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